I recognized that ω is -dθ/dt since it is in the opposite direction. This shows that the magnitude of a is rω2. If I were to draw the vector on the graph, it would also show that a is pointing toward the hinge.

But how would I show that a is always pointing toward the hinge?

Is this argument valid:

Let q be the a vector from the tip to the center, then

qx = sin(θ)
qy = -cos(θ)

Now, a • q = rω2cos(ф) = rω2, where ф is the angle between the two vectors and ф=0 in this case. Therefore a and q are in the same direction.